Question

A system of two identical rods (L-shaped) of mass m and length l are resting on a peg P as shown in the figure. If the system is displaced in its plane by a small angle θ , find the period of oscillations.

a ) 2 π 2 l 3 g b ) 2 π 2 2 l 3 g c ) 2 π 2 l 3 g d ) 3 π l 3 g

Answer 1

time period of oscillation is given by,
$T=2\pi\sqrt{\frac{I_{\textbf{support}}}{mgl_{\textbf{centre of mass}}}}$

where $I_{\textbf{support}}$ is moment of inertia of rods about supporting point.
e.g., $I_{\textbf{support}}$ = 2 × momentum of inertia of rod about one end
= 2ml²/3

$l_{\textbf{centre of mass}}$ is the distance of rod from the centre of mass.
e.g., $l_{\textbf{centre of mass}}$ = lcos45° [ as both rods inclined by 90° , centre of mass must lies bisector line of 90° ]
so, $l_{\textbf{centre of mass}}$ = l/√2

now, $T=2\pi\sqrt{\frac{2\sqrt{2}ml^2}{3mgl}}$

hence, time period, T = $2\pi\sqrt{\frac{2\sqrt{2}l}{3g}}$